Why is math so effective at describing the universe? What is reality is made of math? What does it mean for us to be conscious beings in a mathematical structure? I discuss these questions and more in today’s Ask a Spaceman!

Support the show: http://www.patreon.com/pmsutter

All episodes: http://www.AskASpaceman.com

Watch on YouTube: http://www.youtube.com/PaulMSutter

Read a book: https://www.pmsutter.com/books

Keep those questions about space, science, astronomy, astrophysics, physics, and cosmology coming to #AskASpaceman for COMPLETE KNOWLEDGE OF TIME AND SPACE!

Big thanks to my top Patreon supporters this month: Justin G, Chris L, Alberto M, Duncan M, Corey D, Michael P, Naila, Sam R, Joshua, Scott M, Rob H, Scott M, Louis M, John W, Alexis, Gilbert M, Rob W, Jessica M, Jules R, Jim L, David S, Scott R, Heather, Mike S, Pete H, Steve S, Lisa R, Kevin B, Aileen G, Steven W, Deb A, Michael J, Phillip L, Steven B, Mark R, Alan B, Craig B, Richard K, Joe R, David P, Justin, Tracy F, Ella F, Thomas K, James C, Syamkumar M, Homer V, Mark D, Bruce A, Tim Z, Linda C, The Tired Jedi, Bob C, Stephen A, James R, Robert O, Allen E, Michael S, Reinaldo A, Sheryl, David W, Chris, Michael S, Erlend A, James D, Karl W, Den K, Edward K, Scott K, Vivek D, Barbara C, Brad, Azra K, Steve R, Koen G, and Scott N!

Hosted by Paul M. Sutter.

 

EPISODE TRANSCRIPT (AUTO-GENERATED)

Imagine you walk into a parking lot full of cars. You have in your pocket one single key. It's the key to your car. The same key you've always used, the same key you've always trusted, the same key that you always manage to realize that you've lost right before you're rushing out the door. But have you ever had one of those moments when you walk up to the wrong car? Maybe it's the same color, the same make and model, and it takes you a minute to realize that something's off about the car. And you stop yourself before you unlock the door and get in. But what if your key unlocked that other car? That'd be kind of weird, right? What are the chances that your key, specifically designed for your car and your car alone, just somehow, by sheer coincidence, unlocked a car very similar to, but not exactly, yours? What if you kept going? What if you tried your key on the next car, and the next, and the next? What if your one key unlocked every single car in the parking lot? It'd seem like magic, wouldn't it? In 1960, the physicist Eugene Wigner penned an essay, The Unreasonable Effectiveness of Mathematics in the Natural Sciences.

In it, he laid out how in the four centuries of scientific exploration of the universe, ever since the days of Galileo, math has been really, really good at doing the job. He said, quote, The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift, which we neither understand nor deserve. Math is a key that unlocks door after door after door. So why is math so good? How are we able to make so much progress? Well, maybe math is more than a way to describe the universe. Maybe it's more than a blueprint. Maybe it's the whole thing. This is called the mathematical universe hypothesis. And in usual Ask a Spaceman fashion, I'm going to do two things. One, I'm going to tell you what I think up front. But then two, I'm also going to try to give a balanced perspective so that you can make up your own mind. And here's what I think, right up front. I don't think the universe is made of math. But you don't have to agree with me. We're all good.

And either way, it's a really fun idea to play with, which is why we're all here in the first place. To get things started, we need to talk about math. For the longest time, at least in the Western tradition, math and physics were different, to the point that physics was called natural philosophy. Math described things that were fixed, like geometry and arithmetic and musical harmonies and, of course, the appearance of the stars and planets. These were all fundamental patterns of the universe that simply were, and we could use numbers and formulas and relations, like the Pythagorean theorem for triangles, to uncover those hidden patterns. Physics, on the other hand, was all about change. It was about the process of evolution and manifestation and movement. Math didn't belong here. After all, there's no fundamental pattern to a baby growing up to become an adult or a dolphin swimming in the ocean or fire burning through a log. That's all change. That's all physics. And then Galileo happened.

Now, Galileo did a lot of things, but one of his biggest contributions to physics was the application of math, which, by the way, offended the physicists at the time that there was this lowly mathematician poking his nose in places where it didn't belong. But that's a story for another day. He used and applied mathematics to understand what he saw in nature. He said, "...philosophy, that is, nature, is written in that great book which ever is before our eyes." But we cannot understand it if we do not first learn the language and grasp the symbols in which it is written. The book is written in mathematical language, and the symbols are triangles, circles, and other geometrical figures, without whose help it is impossible to comprehend a single word of it, without which one wanders in vain through a dark labyrinth. One of Galileo's first applications of mathematics to the study of nature was with a pendulum. What's amazing is that pendulums have been around since basically forever, but Galileo was the first to notice that the period of the swing did not depend on the weight of the mass at the end of the rope.

It only depended on the length. And he noticed that big swings took just as much time as little swings. He discovered a deep pattern in nature, the motion of a pendulum, described with mathematics, and used that for all sorts of useful and interesting applications, like making reliable clocks. And from there, the game of modern physics really took off. After Galileo, you have people like Johannes Kepler, you know I can't resist bringing him awkwardly into any conversation, who discovered a literal geometric figure in the orbits of the planets, that figure being ellipses, that figure being ellipticals. And it all comes to a head of century after Galileo with Isaac Newton. Remember, the whole objection to the application of math to physics was that math couldn't capture change. It was all triangle this and equilateral that. Newton was so determined to apply math to his exploration of the universe that he invented a whole new math, the calculus, to be able to capture change and evolution in physical systems.

With the calculus, the whole universe opened up. Everywhere we looked, we found new patterns. They weren't simple geometric figures or algebraic relationships. They were complex, often subtle, and almost always hidden underneath mountains of data. But they were there. And the more comfortable we got with applying math to understanding the universe, the more mathematical tools we invented, or stole from mathematicians when they weren't looking, to search even deeper. Today, physics is a mathematical exploration of the universe. We look for patterns, structures, symmetries, and relationships. We use math to capture and describe those patterns and structures and symmetries and relationships. We take properties of objects that we can measure or identify, like mass or velocity or electric charge, and we find how these properties relate to each other and influence behavior. For example, I can identify a property. Let's call it mass. And I suppose that the mass of an object can influence other objects.

I call that influence the gravitational force. And I can write down an equation, a simple relationship, that tells me how much mass leads to how much gravitational force. And I have another equation to tell me how the gravitational force makes other objects move. Galileo found the key. And for four centuries, we've been opening door after door after door in the process completely rewriting our understanding of the physical universe. Our perspectives, our views of how the universe works are radically different than they were prior to Galileo and prior to the application of math. The universe is chaotic and messy and almost incomprehensible. But there are patterns. There are relationships. There are repeatable, predictable behaviors that we can identify from the orbits of the planets to the Higgs boson. Mathematics is perfectly suited to describe all that. Maybe a little too perfectly suited. Like, it shouldn't be this easy. Yeah, I know, physics is kind of hard, and it's taken us centuries to reach our present level of knowledge, and we've still got a long way to go from complete knowledge of time and space.

But on the other hand, look at all the technological marvels that fill our lives. The smartphones, and GPS satellites, and cures for diseases. We have all that stuff because of science, and science works because math is just so dang handy at describing the universe. We did natural philosophy for thousands of years, and we made progress for sure. But once we started using math, it seemed like there were new revelations every single day. Because there were. Why is math so good? Well, maybe math isn't just a description of nature. Maybe math is nature. And the reason it's so good at its job is because we finally, after millennia of attempts, have hit upon the secret language of nature, just like Galileo said. Maybe the universe is made of math. Now, this is an old idea, but the latest version of it comes to us from cosmologist Max Tegmark. I wouldn't exactly call him a friend of mine. We don't hang out often enough, but definitely somewhere on the colleague acquaintance spectrum. Anyway, back in 2014, Max wrote a book, Our Mathematical Universe, which outlined this philosophy.

Now, the line between physics and metaphysics and philosophy is often blurry. In the book, Max claims that his idea is physics, that he's making a testable prediction. To me, and I know I'm showing my bias here, so please judge for yourself when I talk about it later, these claims aren't all that strong, which means I would put the following discussion firmly in the metaphysics category. Which is fine. There's nothing wrong with metaphysics. Philosophy is useful and important and worth debating and exploring. So let's dig in to what old Max Tegmark has to say about the universe. Let's start with an assumption. It's a big one and a debatable one, but we're going to say it out loud and move on. That there is an external, objective reality. As in, we're not making up reality in our own minds, and it exists independently of us conscious observers. The whole process of science is to discover and explain that external objective reality, and we've been using math very successfully to do just that.

Now, like I said, you can debate that if you want. We're not going to today, but feel free to do so on your own time. But there's way more to science than math. There are words and concepts and ideas that are very, very human. We might have some beautiful mathematical theory like general relativity or quantum mechanics. But then we also have a bunch of human-concocted concepts to wrap around the math, like wave function and space-time and equivalence principle, and even mass and charge and force. Tegmark argues that this is all baggage. It's a layer of our subjective human-centered view of the universe that sits on top of the real deal, the math itself. So we need to apply Occam's razor. We need to make our perspectives as simple as possible. We need to dump all the baggage overboard because it's just getting in the way. The real structure of reality that's objective, that exists independently of us, will not have any human-centered baggage. That's all created by us humans. Actual reality is baggage-free.

It's streamlined. It's efficient. It's straightforward. And what happens when you strip out all the baggage from our theories of physics? You're left with bare, raw mathematics. This isn't Occam's razor. It's more like Occam's sledgehammer. Tegmark is saying that math is more than a useful tool to study the universe. He's saying that once we get down to brass tacks, it's just math. Take a chair. Strip away the baggage, the color, the mass, the atoms, the forces, all the human words we would use to describe that chair. Once you remove all of those human-derived concepts, you're left with relationships, symmetries, structures. Yes, we have equations to describe those relationships and symmetries and structures, but that's just a description. What you're actually left with at the end of the day is math. Math is the universe and the universe is math. There is no distinction. There's no difference between them. If we work at physics hard enough, then math won't just reveal the universe to us.

Instead, the universe will be revealed as even more math. Some physicists are on a hunt for a theory of everything, a single unified theory that describes all the forces of nature. Tegmark says that if he's right, then this theory of everything wouldn't just stop there. It would also explain all the different kinds of particles, all their possible interactions, and all the properties of the universe. You know how I did that episode a while ago on the constants of nature? Yeah, a proper theory of everything wouldn't have any constants, no speed of light, no charge on the electron, not even the number of space and time dimensions. It would be a single equation, okay, maybe a set of equations, that explains all of reality, including itself, which is kind of wild to think about if you're into that sort of thing. A proper theory of everything would also explain contributing to Patreon. That's patreon.com slash P-M-S-U-T-T-E-R. Go there now to continue supporting the show. I truly do appreciate all of your contributions.

A theory of everything would literally be a theory of everything. And because that single mathematical equation could describe all of reality, why don't we just cut out the middleman and say that the mathematical equation is all of reality? I mean, Occam's razor, right? Why make things more complicated? Check this out. Beginning in the 1980s, another physicist, Roger Penrose, who honestly we haven't met enough in this series, so feel free to ask about him, came up with what he called the Triangle of Reality, which sounds like the base idea of the nerdiest cult in history. And later when I get to talk about the Pythagoreans, you'll see that I'm right. The three corners of Penrose's Triangle of Reality. are math, matter, and mind. We have math, which appears to be a set of objective, abstract truths like 2 plus 2 equals 4 kind of stuff. We have our mind, which is our subjective experience of the world. And we have matter, which is, well, the world. So which is responsible for which? Do our minds create math? Does matter create mind? How does it all work? The mathematical universe concepts breaks the triangle.

It says it's just math. Math is the universe. Math is matter. And that means math is mind. According to the mathematical universe hypothesis, we, you, me, every conscious being, are just equations, relationships, logical structures. There's nothing more to us. There's no flesh and bone here. There's not even our own thought process. What we think of as our conscious, subjective experience of the world, like growing older, enjoying a slice of cheese, having that awkward moment when you miss a high five, it's all an illusion. It's all just math. It's all just a structure that never changes and never evolves. For an example, consider the Mandelbrot set. It's a fractal. You zoom in on any one part of it and you just get more of the same repeating patterns. It's all governed by an extremely simple equation, but it contains an enormous amount of complexity inside of it. Imagine writing down an equation that creates a fractal so complex, so intricate, that if you zoom down into it far enough, you would see an extremely complicated pattern emerge, a structure that is so twisted up on itself that it is aware of itself and aware of its environment.

Tegmark calls this the difference between the bird perspective and the frog perspective. From the viewpoint of the bird, we can see the whole complex manifestation of the math. We can see these self-aware substructures, just like we can see a beautiful pattern appear in the Mandelbrot set. But we, inside of it, don't get the perspective. We're in the math because we are math. We're made of math. So we're just frogs. We can't zoom out to see how our little corner of the mathematical pattern is just one structure amongst many. We think we have our minds in the universe around us. Yes, this means there is no free will. There's not even past and future. There's no passage of time. There's not even space. There's just math. Just equations. Just a set of logical, orderly relationships that exist. The pattern of your life already exists. If you were a bird, you could rise above and see your whole life played out. I'm going to use an analogy, and I have to admit it's a bad one, but I'm using it to illustrate just how wild this idea is.

So take my hand, because we're going right into crazy town. It's like we're characters in a video game. From our perspective, the world exists, we exist, stuff exists, we talk and laugh and move around, but it's all just software. It's just logic and programming. There is no us. There's just bits of code. This analogy is bad because you immediately think of either A, the matrix, or B, the simulation argument. That's because for this analogy to work, you don't just need the software, you need the hardware. You need the computer. In the mathematical universe, there is no hardware. There is no computer. There's just code. Here's a slightly better analogy. It's more like we're the characters in a story. From our perspective, the story unfolds page by page, and we never know what comes next. But the book is already written, found, and printed. It's done. It exists. Yes, we are real, just like the characters in a book are real, but we're really just words on a page. 2 plus 2 equals 4. You can design a computer program to compute this.

You can shove a bunch of rocks together to demonstrate this. But it's still true. 2 plus 2 equals 4, even in a world without computers and without rocks. The mathematical structure doesn't need a substrate. It doesn't need an engine to exist. The math still exists. That's heavy stuff, man. Like I said, it's fun to think about. But as we start to think about it, some questions come up, which is also kind of why this kind of stuff is fun to think about. And the first question that comes to mind is, Hey, isn't there a lot of math out there? Like, a lot, a lot? There is a vast array of potential logical systems that could describe all sorts of potential universes. Like universes with five spatial dimensions, or two time dimensions, or 47 forces, or 15 versions of the electron, or a speed of light that's half of what we experience, and on and on and on and on and on. Why should we have this universe with this kind of structure? Tag Mark has an answer for you. And that answer is second verse same as the first, the multiverse.

That's right. There's a universe for everybody. Every possible mathematical structure is realized and existing all in parallel. We just happen to occupy this one. We are self-aware little subsets in a larger mathematical framework because it's the kind that allows sentient beings to arise. Hey, isn't that just the anthropic principle that the universe is the way it is because if it wasn't the way it is, then we wouldn't be here to see it? Why, yes, it is. And the second question comes to us from none other than the final boss of mathematics himself, Kurt Gödel. Now, Gödel didn't actually critique Tegmark's mathematical universe because Gödel has been dead for a long time. But in the early 20th century, he did something remarkable. He showed that for any sufficiently complex formal system, mathematics, there are statements that are true but cannot be proven. I am definitely not going to get into the weeds of this one, but hey, I'm up for the challenge if anyone wants to ask about Gödel's incompleteness theorem.

The point that matters for us is that there is no mathematical system that is both complex and complete. Yes, there are many, many technicalities that I'm glossing over, so deal with it, math nerds. We just need to move on. point here is, look, buddy, you want the universe to be made of math. That's cute and all. But there's no mathematical system that is complex enough to have a universe and be complete. So how can you have a complete universe that actually functions as, you know, a universe? Tag Mark has an answer for you, too. He says maybe not all possible mathematical structures make up the multiverse, but only a small subset of mathematical structures that are computable. In other words, the math that is our universe isn't complex enough to run into any Kurt Gödel problems. We're just actually just complex enough to have, like, sentience, but simple enough to escape any problems with Gödel's incompleteness theorem. Ta-da! A universe made of math that answers some major criticisms.

We cool? Well, kind of. Like I said at the beginning, I'm not really keen on the idea of the mathematical universe. My own personal biggest objection stems from the whole point of Occam's razor. Make things as simple as possible. And that sounds nice as you're removing baggage to arrive at a universe made of math, but then to explain reality, which is a rather large undertaking, you have to start adding things back in, like a multiverse, like an anthropic principle, like the restriction to simple enough mathematical structures. I don't know, that seems to me like baggage, but of a different kind. Besides, simple doesn't always mean correct. But old Willy Ockham proposed his razor as a guide to approaching thinking, not as a be-all, end-all decider of rightness. Look, physics in the 19th century was a heck of a lot simpler than physics in the 21st century. We had gravity, electromagnetism, and, like, heat. That was it. Now we have quantum this and superconducting that, and oh, hello, there's the Higgs boson.

Is 19th century physics correct because it's simpler? I sure hope not. There are plenty of other objections to the mathematical universe. Here's one. We can concoct all sorts of weird particles, forces, properties of nature. Things that violate no known laws of physics. Things that by all rights should exist in the universe, but don't. But if the universe is made of math, why don't these things exist even when they could? Who gets to decide what's in our corner of the multiverse and what's not? And hey, let's go back to Godel. Us humans can see above the map. We can point to statements and say, oh, wow, that's true, even though the mathematical system itself has no way of proving it within its own structures. So if we're just made of math, and if Tegmark is right, made of the simpler kind of math that avoids paradoxes and whatnot, how are we able to see above the system that we're made of? Shouldn't the math that is our universe also be responsible for all the math that we come up with? How can the math that we write down be more complex than the math that the universe is made of? We're literally thinking about mathematical systems that are more complex than necessary to create a universe.

Tegmark says that we can debate this all day long. And we will, thank you very much. But in the end, it comes down to evidence. He argues that if we do find a theory of everything, then that's a clue that we live in a mathematical universe. I mean, I suppose, but also it could mean a bunch of other things, which is why this isn't that much of a physical theory. In the end, whether you think this is a valid approach or not comes down to what you think of math. Not like your personal opinion of it based on fifth grade multiplication tables, but how real you think math is. 2,500 years ago in ancient Greece, the cult of Pythagoras believed that all is number. That by studying geometry and arithmetic, they could arrive at divine truth. Things like the Pythagorean theorem were studied, no, worshipped as mystical revelations. Like I said, the nerdiest cult in history. People today still debate whether math is discovered or if it's invented. If it's discovered, then math already exists. 2 plus 2 equals 4, whether or not there are any rocks hanging around or people to count them.

The circumference of a circle is still 2 pi times its radius, whether or not there are any actual circles around. If you believe that math already exists, then we're just uncovering pieces of that hidden reality. Then you might be willing to believe that math itself is reality. but maybe math is invented. Tegmar calls so much of our subjective experience of the world baggage, but maybe math is baggage too. Maybe we created it to solve real-world problems like figuring out when the river would flood again or how many seeds to plant this year or how much I should trade my cow for. And maybe the fact that math works so well is an illusion of its own. If you wear math-colored glasses, then the universe looks a lot like math. But there's so much to reality that can't be explained by physics, math, or even science, that even if we did have a theory of everything, we couldn't use it to describe the sensation of the color orange or the feeling of grief. Check this quote out from Erwin Schrodinger.

I am very astonished that the scientific picture of the real world around me is very deficient. It gives us a lot of factual information, puts all of our experience in a magnificently consistent order, but it is ghastly silent about all and sundry that is really near to our heart that really matters to us. It cannot tell us a word about red and blue, bitter and sweet, physical pain and physical delight. It knows nothing of beautiful and ugly, good or bad, God and eternity. Science sometimes pretends to answer questions in these domains, but the answers are very often so silly that we are not inclined to take them seriously. That's Schrodinger's opinion. Other people, like Roger Penrose, have very different opinions. They think we can describe the entire universe by math, that if we did have a theory of everything, we could describe all of human experience. Different folks have different takes on these kinds of questions. And whether you like the mathematical universe or not, these questions are very powerful because they challenge us to ask, well, what the heck it is we think we're doing here? Let me leave you with one more incomplete metaphor.

Does Hamlet exist? Sure, yes, of course, you can download a copy of it right now. But if you destroyed all copies and burned every page, would it still exist? What if Shakespeare never happened? What if he never wrote it down? Does Hamlet still exist in the space of all possible combinations of words? The answer is yeah, kinda, sorta, but also not really. It took someone like Shakespeare to pluck that particular combination out and make them real in a different way. Or maybe it does exist. And the story of Hamlet is out there and it was just the act of capturing it that made it special. But it still exists in its own way. Stephen Hawking said it best when he asked, What is it that breathes fire into the equations and makes a universe for them to describe? Maybe there is no fire. Maybe there is. Maybe it's God. Maybe it's you and me. My answer? Maybe just in the asking, we're doing everything that the universe requires of us. Thank you to Ed A., Barely Wise, at Al McClintock, Patrick M., Pierre P., Callan, Renee S., and Justin Z.

For the questions that led to today's episode, keep those questions coming to askaspaceman at gmail.com or the website askaspaceman.com. Please keep sharing this series with anyone you know, even people that don't like science. You never know. Leave a review on your favorite podcasting platform that really helps to show visibility. Of course, if you can, I truly do appreciate any contributions to the show. That's patreon.com slash pmsutter. I'd like to thank my top Patreon contributors this month. They are Justin G., Chris L., Alberto M., Duncan M., Corey D., Michael P., Nylas M., R., Joshua Scott M., Rob H., Scott M., Lewis M., John W., Alexis Gilbert M., Rob W., Jessica M., Jules R., Jim L., David S., Scott R., Heather M., My guests, Pete H., Steve S., Lisa R., Kevin B., Eileen G., Stephen W., Deb A., Michael J., Philip L., and Stephen B. Thank you so much for listening, and I'll see you next time for more Complete Knowledge of Time and Space.

Comment